Fiolhais c et al (eds) a primer in density functional theory (lnp 620 2003)(isbn 3540030832)(277s)

1 Density Functionals for Non-relativistic
Coulomb Systems in the New Century
John P. Perdew∗ and Stefan Kurth†
∗

Department of Physics and
Quantum Theory Group, Tulane University,
New Orleans LA 70118, USA

†

Institut f¨
ur Theoretische Physik,
Freie Universit¨
at
Berlin, Arnimallee 14, 14195 Berlin, Germany

John Perdew

1.1
1.1.1

Introduction
Quantum Mechanical Many-Electron Problem

The material world of everyday experience, as studied by chemistry and condensed-matter physics, is built up from electrons and a few (or at most a few
hundred) kinds of nuclei . The basic interaction is electrostatic or Coulombic: An electron at position r is attracted to a nucleus of charge Z at R by
the potential energy −Z/|r − R|, a pair of electrons at r and r repel one
another by the potential energy 1/|r − r |, and two nuclei at R and R repel
one another as Z Z/|R − R |. The electrons must be described by quantum
mechanics, while the more massive nuclei can sometimes be regarded as classical particles. All of the electrons in the lighter elements, and the chemically

important valence electrons in most elements, move at speeds much less than
the speed of light, and so are non-relativistic.
In essence, that is the simple story of practically everything. But there
is still a long path from these general principles to theoretical prediction of
the structures and properties of atoms, molecules, and solids, and eventually
to the design of new chemicals or materials. If we restrict our focus to the
important class of ground-state properties, we can take a shortcut through
density functional theory.
These lectures present an introduction to density functionals for nonrelativistic Coulomb systems. The reader is assumed to have a working knowledge of quantum mechanics at the level of one-particle wavefunctions ψ(r) [1].
The many-electron wavefunction Ψ (r1 , r2 , . . . , rN ) [2] is briefly introduced
here, and then replaced as basic variable by the electron density n(r). Various
terms of the total energy are defined as functionals of the electron density, and
some formal properties of these functionals are discussed. The most widelyused density functionals – the local spin density and generalized gradient
C. Fiolhais, F. Nogueira, M. Marques (Eds.): LNP 620, pp. 1–55, 2003.
c Springer-Verlag Berlin Heidelberg 2003


2

John P. Perdew and Stefan Kurth

approximations – are then introduced and discussed. At the end, the reader
should be prepared to approach the broad literature of quantum chemistry
and condensed-matter physics in which these density functionals are applied
to predict diverse properties: the shapes and sizes of molecules, the crystal structures of solids, binding or atomization energies, ionization energies
and electron affinities, the heights of energy barriers to various processes,
static response functions, vibrational frequencies of nuclei, etc. Moreover,
the reader’s approach will be an informed and discerning one, based upon
an understanding of where these functionals come from, why they work, and
how they work.

These lectures are intended to teach at the introductory level, and not
to serve as a comprehensive treatise. The reader who wants more can go to
several excellent general sources [3,4,5] or to the original literature. Atomic
units (in which all electromagnetic equations are written in cgs form, and
the fundamental constants , e2 , and m are set to unity) have been used
throughout.
1.1.2

Summary of Kohn–Sham Spin-Density Functional Theory

This introduction closes with a brief presentation of the Kohn-Sham [6]
spin-density functional method, the most widely-used method of electronicstructure calculation in condensed-matter physics and one of the most widelyused methods in quantum chemistry. We seek the ground-state total energy
E and spin densities n↑ (r), n↓ (r) for a collection of N electrons interacting
with one another and with an external potential v(r) (due to the nuclei in
most practical cases). These are found by the selfconsistent solution of an
auxiliary (fictitious) one-electron Schr¨
odinger equation:
1
σ
− ∇2 + v(r) + u([n]; r) + vxc
([n↑ , n↓ ]; r) ψασ (r) = εασ ψασ (r) ,
2
θ(µ − εασ )|ψασ (r)|2 .

nσ (r) =

(1.1)
(1.2)

α

Here σ =↑ or ↓ is the z-component of spin, and α stands for the set of
remaining one-electron quantum numbers. The effective potential includes a
classical Hartree potential
u([n]; r) =

d3 r

n(r )
,
|r − r |

n(r) = n↑ (r) + n↓ (r) ,

(1.3)
(1.4)

σ
and vxc
([n↑ , n↓ ]; r), a multiplicative spin-dependent exchange-correlation potential which is a functional of the spin densities. The step function θ(µ−εασ )
in (1.2) ensures that all Kohn-Sham spin orbitals with εασ < µ are singly


1

Density Functionals for Non-relativistic Coulomb Systems

3

occupied, and those with εασ > µ are empty. The chemical potential µ is

chosen to satisfy
d3 r n(r) = N .

(1.5)

Because (1.1) and (1.2) are interlinked, they can only be solved by iteration
to selfconsistency.
The total energy is
E = Ts [n↑ , n↓ ] +

d3 r n(r)v(r) + U [n] + Exc [n↑ , n↓ ] ,

(1.6)

Ts [n↑ , n↓ ] =

1
θ(µ − εασ ) ψασ | − ∇2 |ψασ
2

(1.7)

where
σ

α

is the non-interacting kinetic energy, a functional of the spin densities because
(as we shall see) the external potential v(r) and hence the Kohn-Sham orbitals
are functionals of the spin densities. In our notation,

ˆ ασ =
ψασ |O|ψ

∗
ˆ ασ (r) .
d3 r ψασ
(r)Oψ

(1.8)

The second term of (1.6) is the interaction of the electrons with the external
potential. The third term of (1.6) is the Hartree electrostatic self-repulsion
of the electron density
U [n] =

1
2

d3 r d3 r

n(r)n(r )
.
|r − r |

(1.9)

The last term of (1.6) is the exchange-correlation energy, whose functional
derivative (as explained later) yields the exchange-correlation potential
σ
vxc

([n↑ , n↓ ]; r) =

δExc
.
δnσ (r)

(1.10)

Not displayed in (1.6), but needed for a system of electrons and nuclei, is the
electrostatic repulsion among the nuclei. Exc is defined to include everything
else omitted from the first three terms of (1.6).
If the exact dependence of Exc upon n↑ and n↓ were known, these equations would predict the exact ground-state energy and spin-densities of a
many-electron system. The forces on the nuclei, and their equilibrium posi∂E
tions, could then be found from − ∂R
.
In practice, the exchange-correlation energy functional must be approximated. The local spin density [6,7] (LSD) approximation has long been popular in solid state physics:
LSD
Exc
[n↑ , n↓ ] =

d3 r n(r)exc (n↑ (r), n↓ (r)) ,

(1.11)


4

John P. Perdew and Stefan Kurth

where exc (n↑ , n↓ ) is the known [8,9,10] exchange-correlation energy per particle for an electron gas of uniform spin densities n↑ , n↓ . More recently, generalized gradient approximations (GGA’s) [11,12,13,14,15,16,17,18,19,20,21]

have become popular in quantum chemistry:
GGA
[n↑ , n↓ ] =
Exc

d3 r f (n↑ , n↓ , ∇n↑ , ∇n↓ ) .

(1.12)

The input exc (n↑ , n↓ ) to LSD is in principle unique, since there is a possible system in which n↑ and n↓ are constant and for which LSD is exact. At least in this sense, there is no unique input f (n↑ , n↓ , ∇n↑ , ∇n↓ ) to
GGA. These lectures will stress a conservative “philosophy of approximation” [20,21], in which we construct a nearly-unique GGA with all the known
correct formal features of LSD, plus others. We will also discuss how to go
beyond GGA.
The equations presented here are really all that we need to do a practical
calculation for a many-electron system. They allow us to draw upon the
intuition and experience we have developed for one-particle systems. The
many-body effects are in U [n] (trivially) and Exc [n↑ , n↓ ] (less trivially), but
we shall also develop an intuitive appreciation for Exc .
While Exc is often a relatively small fraction of the total energy of an
atom, molecule, or solid (minus the work needed to break up the system
into separated electrons and nuclei), the contribution from Exc is typically
about 100% or more of the chemical bonding or atomization energy (the work
needed to break up the system into separated neutral atoms). Exc is a kind of
“glue”, without which atoms would bond weakly if at all. Thus, accurate approximations to Exc are essential to the whole enterprise of density functional
theory. Table 1.1 shows the typical relative errors we find from selfconsistent
calculations within the LSD or GGA approximations of (1.11) and (1.12).
Table 1.2 shows the mean absolute errors in the atomization energies of 20
molecules when calculated by LSD, by GGA, and in the Hartree-Fock approximation. Hartree-Fock treats exchange exactly, but neglects correlation
completely. While the Hartree-Fock total energy is an upper bound to the
true ground-state total energy, the LSD and GGA energies are not.

In most cases we are only interested in small total-energy changes associated with re-arrangements of the outer or valence electrons, to which the
inner or core electrons of the atoms do not contribute. In these cases, we
can replace each core by the pseudopotential [22] it presents to the valence
electrons, and then expand the valence-electron orbitals in an economical
and convenient basis of plane waves. Pseudopotentials are routinely combined with density functionals. Although the most realistic pseudopotentials
are nonlocal operators and not simply local or multiplication operators, and
although density functional theory in principle requires a local external potential, this inconsistency does not seem to cause any practical difficulties.
There are empirical versions of LSD and GGA, but these lectures will
only discuss non-empirical versions. If every electronic-structure calculation


1

Density Functionals for Non-relativistic Coulomb Systems

5

Table 1.1. Typical errors for atoms, molecules, and solids from selfconsistent KohnSham calculations within the LSD and GGA approximations of (1.11) and (1.12).
Note that there is typically some cancellation of errors between the exchange (Ex )
and correlation (Ec ) contributions to Exc . The “energy barrier” is the barrier to a
chemical reaction that arises at a highly-bonded intermediate state
Property

LSD

GGA

Ex
Ec
bond length

structure
energy barrier

5% (not negative enough)
100% (too negative)
1% (too short)
overly favors close packing
100% (too low)

0.5%
5%
1% (too long)
more correct
30% (too low)

Table 1.2. Mean absolute error of the atomization energies for 20 molecules, evaluated by various approximations. (1 hartree = 27.21 eV) (From [20])
Approximation

Mean absolute error (eV)

Unrestricted Hartree-Fock
LSD
GGA
Desired “chemical accuracy”

3.1 (underbinding)
1.3 (overbinding)
0.3 (mostly overbinding)
0.05

were done at least twice, once with nonempirical LSD and once with nonempirical GGA, the results would be useful not only to those interested in the
systems under consideration but also to those interested in the development
and understanding of density functionals.

1.2
1.2.1

Wavefunction Theory
Wavefunctions and Their Interpretation

We begin with a brief review of one-particle quantum mechanics [1]. An
electron has spin s = 12 and z-component of spin σ = + 12 (↑) or − 12 (↓).
The Hamiltonian or energy operator for one electron in the presence of an
external potential v(r) is
ˆ = − 1 ∇2 + v(r) .
h
2

(1.13)

The energy eigenstates ψα (r, σ) and eigenvalues εα are solutions of the timeindependent Schr¨
odinger equation
ˆ α (r, σ) = εα ψα (r, σ) ,
hψ

(1.14)


6

John P. Perdew and Stefan Kurth

and |ψα (r, σ)|2 d3 r is the probability to find the electron with spin σ in volume
element d3 r at r, given that it is in energy eigenstate ψα . Thus
d3 r |ψα (r, σ)|2 = ψ|ψ = 1 .

(1.15)

σ

ˆ commutes with sˆz , we can choose the ψα to be eigenstates of sˆz , i.e.,
Since h
we can choose σ =↑ or ↓ as a one-electron quantum number.
The Hamiltonian for N electrons in the presence of an external potential
v(r) is [2]
ˆ = −1
H
2

N

∇2i +

i=1

N

v(ri ) +
i=1

1
2

i

j=i

1
|ri − rj |

= Tˆ + Vˆext + Vˆee .

(1.16)

The electron-electron repulsion Vˆee sums over distinct pairs of different elecˆ
trons. The states of well-defined energy are the eigenstates of H:
ˆ k (r1 σ1 , . . . , rN σN ) = Ek Ψk (r1 σ1 , . . . , rN σN ) ,
HΨ

(1.17)

where k is a complete set of many-electron quantum numbers; we shall be
interested mainly in the ground state or state of lowest energy, the zerotemperature equilibrium state for the electrons.
Because electrons are fermions, the only physical solutions of (1.17) are
those wavefunctions that are antisymmetric [2] under exchange of two electron labels i and j:
Ψ (r1 σ1 , . . . , ri σi , . . . , rj σj , . . . , rN σN ) =
− Ψ (r1 σ1 , . . . , rj σj , . . . , ri σi , . . . , rN σN ) .

(1.18)

There are N ! distinct permutations of the labels 1, 2, . . . , N, which by (1.18)
all have the same |Ψ |2 . Thus N ! |Ψ (r1 σ1 , . . . , rN σN )|2 d3 r1 . . . d3 rN is the
probability to find any electron with spin σ1 in volume element d3 r1 , etc.,
and
1
d3 r1 . . . d3 rN N ! |Ψ (r1 σ1 , . . . , rN σN )|2 = |Ψ |2 = Ψ |Ψ = 1 .
N ! σ ...σ
1

N

(1.19)
We define the electron spin density nσ (r) so that nσ (r)d3 r is the probability to find an electron with spin σ in volume element d3 r at r. We find nσ (r)
by integrating over the coordinates and spins of the (N − 1) other electrons,
i.e.,
nσ (r) =

1
(N − 1)! σ

d3 r2 . . .

d3 rN N !|Ψ (rσ, r2 σ2 , . . . , rN σN )|2

2 ...σN

d3 r2 . . .

=N
σ2 ...σN

d3 rN |Ψ (rσ, r2 σ2 , . . . , rN σN )|2 .

(1.20)


1

Density Functionals for Non-relativistic Coulomb Systems

7

Equations (1.19) and (1.20) yield
d3 r nσ (r) = N .

(1.21)

σ

Based on the probability interpretation of nσ (r), we might have expected the
right hand side of (1.21) to be 1, but that is wrong; the sum of probabilities
of all mutually-exclusive events equals 1, but finding an electron at r does not
exclude the possibility of finding one at r , except in a one-electron system.
Equation (1.21) shows that nσ (r)d3 r is the average number of electrons of
spin σ in volume element d3 r. Moreover, the expectation value of the external
potential is
Vˆext = Ψ |

N

v(ri )|Ψ =

d3 r n(r)v(r) ,

(1.22)

i=1

with the electron density n(r) given by (1.4).
1.2.2

Wavefunctions for Non-interacting Electrons

As an important special case, consider the Hamiltonian for N non-interacting
electrons:
N
1
ˆ non =
− ∇2i + v(ri ) .
(1.23)
H
2
i=1
The eigenfunctions of the one-electron problem of (1.13) and (1.14) are spin
orbitals which can be used to construct the antisymmetric eigenfunctions Φ
ˆ non :
of H
ˆ non Φ = Enon Φ .
H
(1.24)

Let i stand for ri , σi and construct the Slater determinant or antisymmetrized
product [2]
1
Φ= √
N!

(−1)P ψα1 (P 1)ψα2 (P 2) . . . ψαN (P N ) ,

(1.25)

P

where the quantum label αi now includes the spin quantum number σ. Here
P is any permutation of the labels 1, 2, . . . , N, and (−1)P equals +1 for an
even permutation and −1 for an odd permutation. The total energy is
Enon = εα1 + εα2 + . . . + εαN ,

(1.26)

and the density is given by the sum of |ψαi (r)|2 . If any αi equals any αj
in (1.25), we find Φ = 0, which is not a normalizable wavefunction. This is
the Pauli exclusion principle: two or more non-interacting electrons may not
occupy the same spin orbital.


8

John P. Perdew and Stefan Kurth

As an example, consider the ground state for the non-interacting helium

atom (N = 2). The occupied spin orbitals are
ψ1 (r, σ) = ψ1s (r)δσ,↑ ,

(1.27)

ψ2 (r, σ) = ψ1s (r)δσ,↓ ,

(1.28)

and the 2-electron Slater determinant is
1 ψ1 (r1 , σ1 ) ψ2 (r1 , σ1 )
Φ(1, 2) = √
2 ψ1 (r2 , σ2 ) ψ2 (r2 , σ2 )
1
= ψ1s (r1 )ψ1s (r2 ) √ (δσ1 ,↑ δσ2 ,↓ − δσ2 ,↑ δσ1 ,↓ ) ,
2

(1.29)

which is symmetric in space but antisymmetric in spin (whence the total spin
is S = 0).
If several different Slater determinants yield the same non-interacting energy Enon , then a linear combination of them will be another antisymmetˆ non . More generally, the Slater-determinant eigenstates of
ric eigenstate of H
ˆ
Hnon define a complete orthonormal basis for expansion of the antisymmetric
ˆ the interacting Hamiltonian of (1.16).
eigenstates of H,
1.2.3

Wavefunction Variational Principle

The Schr¨
odinger equation (1.17) is equivalent to a wavefunction variational
ˆ
principle [2]: Extremize Ψ |H|Ψ
subject to the constraint Ψ |Ψ = 1, i.e., set
the following first variation to zero:
δ

ˆ
Ψ |H|Ψ
/ Ψ |Ψ

=0.

(1.30)

The ground state energy and wavefunction are found by minimizing the expression in curly brackets.
The Rayleigh-Ritz method finds the extrema or the minimum in a restricted space of wavefunctions. For example, the Hartree-Fock approximation
to the ground-state wavefunction is the single Slater determinant Φ that minˆ / Φ|Φ . The configuration-interaction ground-state wavefuncimizes Φ|H|Φ
tion [23] is an energy-minimizing linear combination of Slater determinants,
restricted to certain kinds of excitations out of a reference determinant. The
Quantum Monte Carlo method typically employs a trial wavefunction which
is a single Slater determinant times a Jastrow pair-correlation factor [24].
Those widely-used many-electron wavefunction methods are both approximate and computationally demanding, especially for large systems where
density functional methods are distinctly more efficient.
The unrestricted solution of (1.30) is equivalent by the method of Lagrange multipliers to the unconstrained solution of
δ

ˆ

Ψ |H|Ψ
− E Ψ |Ψ

=0,

(1.31)


1

Density Functionals for Non-relativistic Coulomb Systems

i.e.,

ˆ − E)|Ψ = 0 .
δΨ |(H

9

(1.32)

Since δΨ is an arbitrary variation, we recover the Schr¨
odinger equation (1.17).
ˆ is an extremum of Ψ |H|Ψ
ˆ
Every eigenstate of H
/ Ψ |Ψ and vice versa.
The wavefunction variational principle implies the Hellmann-Feynman
and virial theorems below and also implies the Hohenberg-Kohn [25] density
functional variational principle to be presented later.

1.2.4

Hellmann–Feynman Theorem

ˆ λ depends upon a parameter λ, and we want to
Often the Hamiltonian H
know how the energy Eλ depends upon this parameter. For any normalized
ˆ λ ), we
variational solution Ψλ (including in particular any eigenstate of H
define
ˆ λ |Ψλ .
Eλ = Ψλ |H
(1.33)
Then

d
dEλ
ˆ λ |Ψλ
=
Ψλ |H
dλ
dλ

λ =λ

+ Ψλ |

ˆλ
∂H
|Ψλ .

∂λ

(1.34)

The first term of (1.34) vanishes by the variational principle, and we find the
Hellmann-Feynman theorem [26]
ˆλ
dEλ
∂H
|Ψλ .
= Ψλ |
∂λ
dλ

(1.35)

Equation (1.35) will be useful later for our understanding of Exc . For now,
we shall use (1.35) to derive the electrostatic force theorem [26]. Let ri be
the position of the i-th electron, and RI the position of the (static) nucleus
I with atomic number ZI . The Hamiltonian
ˆ =
H

N

1
− ∇2i +
2
i=1

i

I

1
−ZI
+
|ri − RI | 2

i

j=i

1
1
+
|ri − rj | 2

I

J=I

ZI ZJ
|RI − RJ |

(1.36)
depends parametrically upon the position RI , so the force on nucleus I is
−

∂E

=
∂RI

=

Ψ −

d3 r n(r)

ˆ
∂H
Ψ
∂RI

ZI (r − RI )
+
|r − RI |3

J=I

ZI ZJ (RI − RJ )
,
|RI − RJ |3

(1.37)

just as classical electrostatics would predict. Equation (1.37) can be used
to find the equilibrium geometries of a molecule or solid by varying all the
RI until the energy is a minimum and −∂E/∂RI = 0. Equation (1.37) also
forms the basis for a possible density functional molecular dynamics, in which

10

John P. Perdew and Stefan Kurth

the nuclei move under these forces by Newton’s second law. In principle, all
we need for either application is an accurate electron density for each set of
nuclear positions.
1.2.5

Virial Theorem

The density scaling relations to be presented in Sect. 1.4, which constitute
important constraints on the density functionals, are rooted in the same
wavefunction scaling that will be used here to derive the virial theorem [26].
ˆ
Let Ψ (r1 , . . . , rN ) be any extremum of Ψ |H|Ψ
over normalized wavefunctions, i.e., any eigenstate or optimized restricted trial wavefunction (where irrelevant spin variables have been suppressed). For any scale parameter γ > 0,
define the uniformly-scaled wavefunction
Ψγ (r1 , . . . , rN ) = γ 3N/2 Ψ (γr1 , . . . , γrN )

(1.38)

Ψγ |Ψγ = Ψ |Ψ = 1 .

(1.39)

and observe that

The density corresponding to the scaled wavefunction is the scaled density
nγ (r) = γ 3 n(γr) ,

(1.40)

which clearly conserves the electron number:
d3 r nγ (r) =

d3 r n(r) = N .

(1.41)

γ > 1 leads to densities nγ (r) that are higher (on average) and more contracted than n(r), while γ < 1 produces densities that are lower and more
expanded.
ˆ = Tˆ + Vˆ under scaling. By definition
Now consider what happens to H
of Ψ ,
d
Ψγ |Tˆ + Vˆ |Ψγ
=0.
(1.42)
dγ
γ=1
But Tˆ is homogeneous of degree -2 in r, so
Ψγ |Tˆ|Ψγ = γ 2 Ψ |Tˆ|Ψ ,

(1.43)

and (1.42) becomes
2 Ψ |Tˆ|Ψ +

or
2 Tˆ −

d
Ψγ |Vˆ |Ψγ
dγ
N

ri ·
i=1

γ=1

=0,

∂ Vˆ
=0.
∂ri

(1.44)

(1.45)


1

Density Functionals for Non-relativistic Coulomb Systems

11

If the potential energy Vˆ is homogeneous of degree n, i.e., if

then

V (γri , . . . , γrN ) = γ n V (ri , . . . , rN ) ,

(1.46)

Ψγ |Vˆ |Ψγ = γ −n Ψ |Vˆ |Ψ ,

(1.47)

and (1.44) becomes simply
2 Ψ |Tˆ|Ψ − n Ψ |Vˆ |Ψ = 0 .

(1.48)

For example, n = −1 for the Hamiltonian of (1.36) in the presence of a
single nucleus, or more generally when the Hellmann-Feynman forces of (1.37)
vanish for the state Ψ .

1.3
1.3.1

Definitions of Density Functionals
Introduction to Density Functionals

The many-electron wavefunction Ψ (r1 σ1 , . . . , rN σN ) contains a great deal of
information – all we could ever have, but more than we usually want. Because
it is a function of many variables, it is not easy to calculate, store, apply or

even think about. Often we want no more than the total energy E (and its
changes), or perhaps also the spin densities n↑ (r) and n↓ (r), for the ground
state. As we shall see, we can formally replace Ψ by the observables n↑ and
n↓ as the basic variational objects.
While a function is a rule which assigns a number f (x) to a number
x, a functional is a rule which assigns a number F [f ] to a function f . For
ˆ
example, h[Ψ ] = Ψ |H|Ψ
is a functional of the trial wavefunction Ψ , given
ˆ U [n] of (1.9) is a functional of the density n(r), as is the
the Hamiltonian H.
local density approximation for the exchange energy:
ExLDA [n] = Ax

d3 r n(r)4/3 .

(1.49)

The functional derivative δF/δn(r) tells us how the functional F [n]
changes under a small variation δn(r):
δF =

d3 r

δF
δn(r)

δn(r) .

For example,

δExLDA = Ax
= Ax

d3 r

[n(r) + δn(r)]4/3 − n(r)4/3

4
d3 r n(r)1/3 δn(r) ,
3

(1.50)


12

John P. Perdew and Stefan Kurth

so

δExLDA
4
= Ax n(r)1/3 .
δn(r)
3

(1.51)

δU [n]
= u([n]; r) ,

δn(r)

(1.52)

Similarly,

where the right hand side is given by (1.3). Functional derivatives of various
orders can be linked through the translational and rotational symmetries of
empty space [27].
1.3.2

Density Variational Principle

We seek a density functional analog of (1.30). Instead of the original derivation of Hohenberg, Kohn and Sham [25,6], which was based upon “reductio ad
absurdum”, we follow the “constrained search” approach of Levy [28], which
is in some respects simpler and more constructive.
Equation (1.30) tells us that the ground state energy can be found by miniˆ
mizing Ψ |H|Ψ
over all normalized, antisymmetric N -particle wavefunctions:
ˆ
E = min Ψ |H|Ψ
.

(1.53)

Ψ

We now separate the minimization of (1.53) into two steps. First we consider
all wavefunctions Ψ which yield a given density n(r), and minimize over those
wavefunctions:

ˆ
min Ψ |H|Ψ
= min Ψ |Tˆ + Vˆee |Ψ +

Ψ →n

Ψ →n

d3 r v(r)n(r) ,

(1.54)

where we have exploited the fact that all wavefunctions that yield the same
n(r) also yield the same Ψ |Vˆext |Ψ . Then we define the universal functional
F [n] = min Ψ |Tˆ + Vˆee |Ψ = Ψnmin |Tˆ + Vˆee |Ψnmin ,
Ψ →n

(1.55)

where Ψnmin is that wavefunction which delivers the minimum for a given n.
Finally we minimize over all N -electron densities n(r):
E = min Ev [n]
n

= min F [n] +
n

d3 r v(r)n(r)

,

(1.56)

where of course v(r) is held fixed during the minimization. The minimizing
density is then the ground-state density.
The constraint of fixed N can be handled formally through introduction
of a Lagrange multiplier µ:
δ F [n] +

d3 r v(r)n(r) − µ

d3 r n(r)

=0,

(1.57)


1

Density Functionals for Non-relativistic Coulomb Systems

13

which is equivalent to the Euler equation
δF
+ v(r) = µ .
δn(r)

(1.58)

µ is to be adjusted until (1.5) is satisfied. Equation (1.58) shows that the
external potential v(r) is uniquely determined by the ground state density
(or by any one of them, if the ground state is degenerate).
The functional F [n] is defined via (1.55) for all densities n(r) which
are “N -representable”, i.e., come from an antisymmetric N -electron wavefunction. We shall discuss the extension from wavefunctions to ensembles in
Sect. 1.4.5. The functional derivative δF/δn(r) is defined via (1.58) for all densities which are “v-representable”, i.e., come from antisymmetric N -electron
ground-state wavefunctions for some choice of external potential v(r).
This formal development requires only the total density of (1.4), and not
the separate spin densities n↑ (r) and n↓ (r). However, it is clear how to get
to a spin-density functional theory: just replace the constraint of fixed n
in (1.54) and subsequent equations by that of fixed n↑ and n↓ . There are two
practical reasons to do so: (1) This extension is required when the external
potential is spin-dependent, i.e., v(r) → vσ (r), as when an external magnetic
field couples to the z-component of electron spin. (If this field also couples to
the current density j(r), then we must resort to a current-density functional
theory.) (2) Even when v(r) is spin-independent, we may be interested in
the physical spin magnetization (e.g., in magnetic materials). (3) Even when
neither (1) nor (2) applies, our local and semi-local approximations (see (1.11)
and (1.12)) typically work better when we use n↑ and n↓ instead of n.
1.3.3

Kohn–Sham Non-interacting System

For a system of non-interacting electrons, Vˆee of (1.16) vanishes so F [n]
of (1.55) reduces to
ˆ min .
Ts [n] = min Ψ |Tˆ|Ψ = Φmin
n |T |Φn
Ψ →n

(1.59)

Although we can search over all antisymmetric N -electron wavefunctions
in (1.59), the minimizing wavefunction Φmin
for a given density will be a nonn
interacting wavefunction (a single Slater determinant or a linear combination
of a few) for some external potential Vˆs such that
δTs
+ vs (r) = µ ,
δn(r)

(1.60)

as in (1.58). In (1.60), the Kohn-Sham potential vs (r) is a functional of n(r). If
there were any difference between µ and µs , the chemical potentials for interacting and non-interacting systems of the same density, it could be absorbed


14

John P. Perdew and Stefan Kurth

into vs (r). We have assumed that n(r) is both interacting and non-interacting
v-representable.
Now we define the exchange-correlation energy Exc [n] by
F [n] = Ts [n] + U [n] + Exc [n] ,

(1.61)

where U [n] is given by (1.9). The Euler equations (1.58) and (1.60) are consistent with one another if and only if

vs (r) = v(r) +

δU [n] δExc [n]
.
+
δn(r)
δn(r)

(1.62)

Thus we have derived the Kohn-Sham method [6] of Sect. 1.1.2.
The Kohn-Sham method treats Ts [n] exactly, leaving only Exc [n] to be
approximated. This makes good sense, for several reasons: (1) Ts [n] is typically a very large part of the energy, while Exc [n] is a smaller part. (2) Ts [n]
is largely responsible for density oscillations of the shell structure and Friedel
types, which are accurately described by the Kohn-Sham method. (3) Exc [n]
is somewhat better suited to the local and semi-local approximations than is
Ts [n], for reasons to be discussed later. The price to be paid for these benefits
is the appearance of orbitals. If we had a very accurate approximation for
Ts directly in terms of n, we could dispense with the orbitals and solve the
Euler equation (1.60) directly for n(r).
The total energy of (1.6) may also be written as
E=

θ(µ − εασ )εασ − U [n] −

d3 r n(r)vxc ([n]; r) + Exc [n] ,

(1.63)

ασ

where the second and third terms on the right hand side simply remove
contributions to the first term which do not belong in the total energy. The
first term on the right of (1.63), the non-interacting energy Enon , is the only
term that appears in the semi-empirical H¨
uckel theory [26]. This first term
includes most of the electronic shell structure effects which arise when Ts [n]
is treated exactly (but not when Ts [n] is treated in a continuum model like
the Thomas-Fermi approximation or the gradient expansion).
1.3.4

Exchange Energy and Correlation Energy

Exc [n] is the sum of distinct exchange and correlation terms:

where [29]

Exc [n] = Ex [n] + Ec [n] ,

(1.64)

min
ˆ
− U [n] .
Ex [n] = Φmin
n |Vee |Φn

(1.65)

is a single Slater determinant, (1.65) is just the usual Fock inteWhen Φmin

n
gral applied to the Kohn-Sham orbitals, i.e., it differs from the Hartree-Fock


1

Density Functionals for Non-relativistic Coulomb Systems

15

exchange energy only to the extent that the Kohn-Sham orbitals differ from
the Hartree-Fock orbitals for a given system or density (in the same way that
Ts [n] differs from the Hartree-Fock kinetic energy). We note that
min
ˆ ˆ
Φmin
= Ts [n] + U [n] + Ex [n] ,
n |T + Vee |Φn

(1.66)

and that, in the one-electron (Vˆee = 0) limit [9],
Ex [n] = −U [n]

(N = 1) .

(1.67)

The correlation energy is
Ec [n] = F [n] − {Ts [n] + U [n] + Ex [n]}

min
ˆ ˆ
= Ψnmin |Tˆ + Vˆee |Ψnmin − Φmin
.
n |T + Vee |Φn

(1.68)

Since Ψnmin is that wavefunction which yields density n and minimizes Tˆ +
Vˆee , (1.68) shows that
(1.69)
Ec [n] ≤ 0 .
Since Φmin
is that wavefunction which yields density n and minimizes Tˆ ,
n
(1.68) shows that Ec [n] is the sum of a positive kinetic energy piece and a
negative potential energy piece. These pieces of Ec contribute respectively
to the first and second terms of the virial theorem, (1.45). Clearly for any
one-electron system [9]
Ec [n] = 0

(N = 1) .

(1.70)

Equations (1.67) and (1.70) show that the exchange-correlation energy
of a one-electron system simply cancels the spurious self-interaction U [n]. In
the same way, the exchange-correlation potential cancels the spurious selfinteraction in the Kohn-Sham potential [9]
δEx [n]
= −u([n]; r)

δn(r)
δEc [n]
=0
δn(r)
Thus
lim

r→∞

1
δExc [n]
=−
δn(r)
r

(N = 1) ,
(N = 1) .

(N = 1) .

(1.71)
(1.72)

(1.73)

The extension of these one-electron results to spin-density functional theory
is straightforward, since a one-electron system is fully spin-polarized.


16

John P. Perdew and Stefan Kurth

1.3.5

Coupling-Constant Integration

The definitions (1.65) and (1.68) are formal ones, and do not provide much
intuitive or physical insight into the exchange and correlation energies, or
much guidance for the approximation of their density functionals. These insights are provided by the coupling-constant integration [30,31,32,33] to be
derived below.
Let us define Ψnmin,λ as that normalized, antisymmetric wavefunction
which yields density n(r) and minimizes the expectation value of Tˆ + λVˆee ,
where we have introduced a non-negative coupling constant λ. When λ = 1,
Ψnmin,λ is Ψnmin , the interacting ground-state wavefunction for density n. When
λ = 0, Ψnmin,λ is Φmin
n , the non-interacting or Kohn-Sham wavefunction for
density n. Varying λ at fixed n(r) amounts to varying the external potential
vλ (r): At λ = 1, vλ (r) is the true external potential, while at λ = 0 it is the
Kohn-Sham effective potential vs (r). We normally assume a smooth, “adiabatic connection” between the interacting and non-interacting ground states
as λ is reduced from 1 to 0.
Now we write (1.64), (1.65) and (1.68) as
Exc [n]
= Ψnmin,λ |Tˆ + λVˆee |Ψnmin,λ
=

1
0

dλ

λ=1

− Ψnmin,λ |Tˆ + λVˆee |Ψnmin,λ

d
Ψ min,λ |Tˆ + λVˆee |Ψnmin,λ − U [n] .
dλ n

λ=0

− U [n]
(1.74)

The Hellmann-Feynman theorem of Sect. 1.2.4 allows us to simplify (1.74)
to
1
Exc [n] =
d λ Ψnmin,λ |Vˆee |Ψnmin,λ − U [n] .
(1.75)
0

Equation (1.75) “looks like” a potential energy; the kinetic energy contribution to Exc has been subsumed by the coupling-constant integration. We
should remember, of course, that only λ = 1 is real or physical. The KohnSham system at λ = 0, and all the intermediate values of λ, are convenient
mathematical fictions.
To make further progress, we need to know how to evaluate the N -electron
expectation value of a sum of one-body operators like Tˆ, or a sum of twobody operators like Vˆee . For this purpose, we introduce one-electron (ρ1 ) and
two-electron (ρ2 ) reduced density matrices [34] :
ρ1 (r σ, rσ) ≡ N

d3 r2 . . .

d3 rN

σ2 ...σN
∗

Ψ (r σ, r2 σ2 , . . . , rN σN ) Ψ (rσ, r2 σ2 , . . . , rN σN ) , (1.76)
ρ2 (r , r)

d3 r3 . . .

≡ N (N − 1)

d3 rN

σ1 ...σN

|Ψ (r σ1 , rσ2 , . . . , rN σN )|2 .

(1.77)


1

Density Functionals for Non-relativistic Coulomb Systems

From (1.20),

nσ (r) = ρ1 (rσ, rσ) .

17

(1.78)

Clearly also
1
Tˆ = −
2

d3 r
σ

1
Vˆee =
2

∂ ∂
·
ρ1 (r σ, rσ)
∂r ∂r

d3 r d3 r

,

(1.79)

r =r

ρ2 (r , r)
.
|r − r |

(1.80)

We interpret the positive number ρ2 (r , r)d3 r d3 r as the joint probability of
finding an electron in volume element d3 r at r , and an electron in d3 r at
r. By standard probability theory, this is the product of the probability of
finding an electron in d3 r (n(r)d3 r) and the conditional probability of finding
an electron in d3 r , given that there is one at r (n2 (r, r )d3 r ):
ρ2 (r , r) = n(r)n2 (r, r ) .

(1.81)

By arguments similar to those used in Sect. 1.2.1, we interpret n2 (r, r ) as
the average density of electrons at r , given that there is an electron at r.
Clearly then
d3 r n2 (r, r ) = N − 1 .

(1.82)

For the wavefunction Ψnmin,λ , we write
n2 (r, r ) = n(r ) + nλxc (r, r ) ,

(1.83)

an equation which defines nλxc (r, r ), the density at r of the exchangecorrelation hole [33] about an electron at r. Equations (1.5) and (1.83) imply
that
d3 r nλxc (r, r ) = −1 ,

(1.84)
which says that, if an electron is definitely at r, it is missing from the rest of
the system.
Because the Coulomb interaction 1/u is singular as u = |r − r | → 0, the
exchange-correlation hole density has a cusp [35,34] around u = 0:
∂
∂u

dΩu λ
n (r, r + u)
4π xc

u=0

= λ n(r) + nλxc (r, r) ,

(1.85)

where dΩu /(4π) is an angular average. This cusp vanishes when λ = 0,
and also in the fully-spin-polarized and low-density limits, in which all other
electrons are excluded from the position of a given electron: nλxc (r, r) = −n(r).
We can now rewrite (1.75) as [33]
Exc [n] =

1
2

d3 r d3 r

n(r)¯

nxc (r, r )
,
|r − r |

(1.86)


18

John P. Perdew and Stefan Kurth

where
n
¯ xc (r, r ) =

1
0

dλ nλxc (r, r )

(1.87)

is the coupling-constant averaged hole density. The exchange-correlation energy is just the electrostatic interaction between each electron and the
coupling-constant-averaged exchange-correlation hole which surrounds it.
The hole is created by three effects: (1) self-interaction correction, a classical
effect which guarantees that an electron cannot interact with itself, (2) the
Pauli exclusion principle, which tends to keep two electrons with parallel
spins apart in space, and (3) the Coulomb repulsion, which tends to keep
any two electrons apart in space. Effects (1) and (2) are responsible for the
exchange energy, which is present even at λ = 0, while effect (3) is responsible

for the correlation energy, and arises only for λ = 0.
If Ψnmin,λ=0 is a single Slater determinant, as it typically is, then the oneand two-electron density matrices at λ = 0 can be constructed explicitly from
the Kohn-Sham spin orbitals ψασ (r):
ρλ=0
(r σ, rσ) =
1

α

∗
θ(µ − εασ )ψασ
(r )ψασ (r) ,

ρλ=0
(r , r) = n(r)n(r ) + n(r)nx (r, r ) ,
2
where
nx (r, r ) = nλ=0
xc (r, r ) = −

σ

|ρλ=0
(r σ, rσ)|2
1
n(r)

(1.88)
(1.89)
(1.90)

is the exact exchange-hole density. Equation (1.90) shows that
nx (r, r ) ≤ 0 ,

(1.91)

so the exact exchange energy
Ex [n] =

1
2

d3 r

d3 r

n(r)nx (r, r )
|r − r |

(1.92)

is also negative, and can be written as the sum of up-spin and down-spin
contributions:
(1.93)
Ex = Ex↑ + Ex↓ < 0 .
Equation (1.84) provides a sum rule for the exchange hole:
d3 r nx (r, r ) = −1 .

(1.94)

Equations (1.90) and (1.78) show that the “on-top” exchange hole density
is [36]
n2↑ (r) + n2↓ (r)
nx (r, r) = −
,
(1.95)
n(r)


1

Density Functionals for Non-relativistic Coulomb Systems

19

which is determined by just the local spin densities at position r – suggesting
a reason why local spin density approximations work better than local density
approximations.
The correlation hole density is defined by
n
¯ xc (r, r ) = nx (r, r ) + n
¯ c (r, r ) ,

(1.96)

and satisfies the sum rule
d3 r n
¯ c (r, r ) = 0 ,

(1.97)

which says that Coulomb repulsion changes the shape of the hole but not
its integral. In fact, this repulsion typically makes the hole deeper but more
short-ranged, with a negative on-top correlation hole density:
n
¯ c (r, r) ≤ 0 .

(1.98)

The positivity of (1.77) is equivalent via (1.81) and (1.83) to the inequality
n
¯ xc (r, r ) ≥ −n(r ) ,

(1.99)

which asserts that the hole cannot take away electrons that were not there
initially. By the sum rule (1.97), the correlation hole density n
¯ c (r, r ) must
have positive as well as negative contributions. Moreover, unlike the exchange
hole density nx (r, r ), the exchange-correlation hole density n
¯ xc (r, r ) can be
positive.
To better understand Exc , we can simplify (1.86) to the “real-space analysis” [37]
n
¯ xc (u)
N ∞
,
(1.100)
Exc [n] =
du 4πu2

2 0
u
where
n
¯ xc (u) =

1
N

d3 r n(r)

dΩu
n
¯ xc (r, r + u)
4π

(1.101)

is the system- and spherical-average of the coupling-constant-averaged hole
density. The sum rule of (1.84) becomes
∞
0

du 4πu2 n
¯ xc (u) = −1 .

(1.102)

As u increases from 0, nx (u) rises analytically like nx (0) +O(u2 ), while
¯ c (0) + O(|u|) as a consequence of the cusp of (1.85).

n
¯ c (u) rises like n
Because of the constraint of (1.102) and because of the factor 1/u in (1.100),
Exc typically becomes more negative as the on-top hole density n
¯ xc (u) gets
more negative.


20

John P. Perdew and Stefan Kurth

1.4

Formal Properties of Functionals

1.4.1

Uniform Coordinate Scaling

The more we know of the exact properties of the density functionals Exc [n]
and Ts [n], the better we shall understand and be able to approximate these
functionals. We start with the behavior of the functionals under a uniform
coordinate scaling of the density, (1.40).
The Hartree electrostatic self-repulsion of the electrons is known exactly
(see (1.9)), and has a simple coordinate scaling:
U [nγ ] =

1
2

=γ

n(γr)n(γr )
|r − r |
n(r1 )n(r1 )
= γU [n] ,
d3 r1
|r1 − r1 |

d3 (γr) d3 (γr )
1
2

d3 r1

(1.103)

where r1 = γr and r1 = γr .
Next consider the non-interacting kinetic energy of (1.59). Scaling all the
wavefunctions Ψ in the constrained search as in (1.38) will scale the density as
in (1.40) and scale each kinetic energy expectation value as in (1.43). Thus
the constrained search for the unscaled density maps into the constrained
search for the scaled density, and [38]
Ts [nγ ] = γ 2 Ts [n] .

(1.104)

We turn now to the exchange energy of (1.65). By the argument of the
min

last paragraph, Φmin
nγ is the scaled version of Φn . Since also
Vˆee (γr1 , . . . , γrN ) = γ −1 Vˆee (r1 , . . . , rN ) ,

(1.105)

and with the help of (1.103), we find [38]
Ex [nγ ] = γ Ex [n] .

(1.106)

In the high-density (γ → ∞) limit, Ts [nγ ] dominates U [nγ ] and Ex [nγ ].
An example would be an ion with a fixed number of electrons N and a
nuclear charge Z which tends to infinity; in this limit, the density and energy
become essentially hydrogenic, and the effects of U and Ex become relatively
negligible. In the low-density (γ → 0) limit, U [nγ ] and Ex [nγ ] dominate
Ts [nγ ].
We can use coordinate scaling relations to fix the form of a local density
approximation
F [n] =

d3 r f (n(r)) .

(1.107)

If F [nλ ] = λp F [n], then
λ−3

d3 (λr) f λ3 n(λr) = λp

d3 r f (n(r)) ,

(1.108)


1

Density Functionals for Non-relativistic Coulomb Systems

21

or f (λ3 n) = λp+3 f (n), whence
f (n) = n1+p/3 .

(1.109)

For the exchange energy of (1.106), p = 1 so (1.107) and (1.109) imply (1.49).
For the non-interacting kinetic energy of (1.104), p = 2 so (1.107) and (1.109)
imply the Thomas-Fermi approximation
d3 r n5/3 (r) .

T0 [n] = As

(1.110)

U [n] of (1.9) is too strongly nonlocal for any local approximation.
While Ts [n], U [n] and Ex [n] have simple scalings, Ec [n] of (1.68) does not.
This is because Ψnmin
, the wavefunction which via (1.55) yields the scaled denγ
sity nγ (r) and minimizes the expectation value of Tˆ + Vˆee , is not the scaled

wavefunction γ 3N/2 Ψnmin (γr1 , . . . , γrN ). The scaled wavefunction yields nγ (r)
but minimizes the expectation value of Tˆ + γ Vˆee , and it is this latter expectation value which scales like γ 2 under wavefunction scaling. Thus [39]
Ec [nγ ] = γ 2 Ec1/γ [n] ,

(1.111)

1/γ

where Ec [n] is the density functional for the correlation energy in a system
for which the electron-electron interaction is not Vˆee but γ −1 Vˆee .
To understand these results, let us assume that the Kohn-Sham non-interacting Hamiltonian has a non-degenerate ground state. In the high-density
limit (γ → ∞), Ψnmin
minimizes just Tˆ and reduces to Φmin
nγ . Now we treat
γ
∆ ≡ Vˆee −

N
i=1

δEx [n]
δU [n]
+
δn(ri )
δn(ri )

(1.112)

as a weak perturbation [40,41] on the Kohn-Sham non-interacting Hamiltonian, and find
| k|∆|0 |2

,
(1.113)
Ec [n] =
E0 − Ek
k=0

where the |k are the eigenfunctions of the Kohn-Sham non-interacting Hamiltonian, and |0 is its ground state. Both the numerator and the denominator
of (1.113) scale like γ 2 , so [42]
lim Ec [nγ ] = constant .

γ→∞

(1.114)

In the low-density limit, Ψnmin
minimizes just Vˆee , and (1.68) then shows
γ
that [43]
Ec [nγ ] ≈ γD[n]
(γ → 0) ,
(1.115)
with an appropriately chosen density functional D[n].


22

John P. Perdew and Stefan Kurth

Generally, we have a scaling inequality [38]
Ec [nγ ] > γEc [n]

(γ > 1) ,

(1.116)

Ec [nγ ] < γEc [n]

(γ < 1) .

(1.117)

If we choose a density n, we can plot Ec [nγ ] versus γ, and compare the result
to the straight line γEc [n]. These two curves will drop away from zero as γ
increases from zero (with different initial slopes), then cross at γ = 1. The
convex Ec [nγ ] will then approach a negative constant as γ → ∞.
1.4.2

Local Lower Bounds

Because of the importance of local and semilocal approximations like (1.11)
and (1.12), bounds on the exact functionals are especially useful when the
bounds are themselves local functionals.
Lieb and Thirring [44] have conjectured that Ts [n] is bounded from below
by the Thomas-Fermi functional
Ts [n] ≥ T0 [n] ,

(1.118)

where T0 [n] is given by (1.110) with
As =

3
(3π 2 )2/3 .
10

(1.119)

We have already established that
λ=1
Ex [n] ≥ Exc [n] ≥ Exc
[n] ,

(1.120)

λ
where the final term of (1.120) is the integrand Exc
[n] of the coupling-constant
integration of (1.75),
λ
Exc
[n] = Ψnmin,λ |Vˆee |Ψnmin,λ − U [n] ,

(1.121)

evaluated at the upper limit λ = 1. Lieb and Oxford [45] have proved that
λ=1
Exc
[n] ≥ 2.273 ExLDA [n] ,

(1.122)

where ExLDA [n] is the local density approximation for the exchange energy,
(1.49), with
3
(1.123)
Ax = −
(3π 2 )1/3 .
4π


1

1.4.3

Density Functionals for Non-relativistic Coulomb Systems

23

Spin Scaling Relations

Spin scaling relations can be used to convert density functionals into spindensity functionals.
For example, the non-interacting kinetic energy is the sum of the separate
kinetic energies of the spin-up and spin-down electrons:
Ts [n↑ , n↓ ] = Ts [n↑ , 0] + Ts [0, n↓ ] .

(1.124)

The corresponding density functional, appropriate to a spin-unpolarized system, is [46]
Ts [n] = Ts [n/2, n/2] = 2Ts [n/2, 0] ,
(1.125)

whence Ts [n/2, 0] = 12 Ts [n] and (1.124) becomes
Ts [n↑ , n↓ ] =

1
1
Ts [2n↑ ] + Ts [2n↓ ] .
2
2

(1.126)

1
1
Ex [2n↑ ] + Ex [2n↓ ] .
2
2

(1.127)

Similarly, (1.93) implies [46]
Ex [n↑ , n↓ ] =

For example, we can start with the local density approximations (1.110) and
(1.49), then apply (1.126) and (1.127) to generate the corresponding local
spin density approximations.
Because two electrons of anti-parallel spin repel one another coulombically, making an important contribution to the correlation energy, there is no
simple spin scaling relation for Ec .
1.4.4

Size Consistency

Common sense tells us that the total energy E and density n(r) for a system,
comprised of two well-separated subsystems with energies E1 and E2 and
densities n1 (r) and n2 (r), must be E = E1 + E2 and n(r) = n1 (r) + n2 (r).
Approximations which satisfy this expectation, such as the LSD of (1.11) or
the GGA of (1.12), are properly size consistent [47]. Size consistency is not
only a principle of physics, it is almost a principle of epistemology: How could
we analyze or understand complex systems, if they could not be separated
into simpler components?
Density functionals which are not size consistent are to be avoided. An
example is the Fermi-Amaldi [48] approximation for the exchange energy,
ExFA [n] = −U [n/N ] ,
where N is given by (1.5), which was constructed to satisfy (1.67).

(1.128)


24

John P. Perdew and Stefan Kurth

1.4.5

Derivative Discontinuity

In Sect. 1.3, our density functionals were defined as constrained searches over
wavefunctions. Because all wavefunctions searched have the same electron
number, there is no way to make a number-nonconserving density variation
δn(r). The functional derivatives are defined only up to an arbitrary constant,
which has no effect on (1.50) when d3 r δn(r) = 0.

To complete the definition of the functional derivatives and of the chemical
potential µ, we extend the constrained search from wavefunctions to ensembles [49,50]. An ensemble or mixed state is a set of wavefunctions or pure
states and their respective probabilities. By including wavefunctions with
different electron numbers in the same ensemble, we can develop a density
functional theory for non-integer particle number. Fractional particle numbers can arise in an open system that shares electrons with its environment,
and in which the electron number fluctuates between integers.
The upshot is that the ground-state energy E(N ) varies linearly between
two adjacent integers, and has a derivative discontinuity at each integer. This
discontinuity arises in part from the exchange-correlation energy (and entirely
so in cases for which the integer does not fall on the boundary of an electronic
shell or subshell, e.g., for N = 6 in the carbon atom but not for N = 10 in
the neon atom).
By Janak’s theorem [51], the highest partly-occupied Kohn-Sham eigenvalue εHO equals ∂E/∂N = µ, and so changes discontinuously [49,50] at an
integer Z:
−IZ (Z − 1 < N < Z)
,
(1.129)
εHO =
−AZ (Z < N < Z + 1)
where IZ is the first ionization energy of the Z-electron system (i.e., the least
energy needed to remove an electron from this system), and AZ is the electron
affinity of the Z-electron system (i.e., AZ = IZ+1 ). If Z does not fall on the
boundary of an electronic shell or subshell, all of the difference between −IZ
and −AZ must arise from a discontinuous jump in the exchange-correlation
potential δExc /δn(r) as the electron number N crosses the integer Z.
Since the asymptotic decay of the density of a finite system with Z electrons is controlled by IZ , we can show that the exchange-correlation potential
tends to zero as |r| → ∞ [52]:
lim

|r|→∞

δExc
=0
δn(r)

(Z − 1 < N < Z) ,

(1.130)

or more precisely
lim

|r|→∞

δExc
1
=−
δn(r)
r

(Z − 1 < N < Z) .

(1.131)

As N increases through the integer Z, δExc /δn(r) jumps up by a positive
additive constant. With further increases in N above Z, this “constant” van-


1

Density Functionals for Non-relativistic Coulomb Systems

25

ishes, first at very large |r| and then at smaller and smaller |r|, until it is all
gone in the limit where N approaches the integer Z + 1 from below.
Simple continuum approximations to Exc [n↑ , n↓ ], such as the LSD
of (1.11) or the GGA of (1.12), miss much or all the derivative discontinuity,
and can at best average over it. For example, the highest occupied orbital
energy for a neutral atom becomes approximately − 12 (IZ + AZ ), the average
of (1.129) from the electron-deficient and electron-rich sides of neutrality. We
must never forget, when we make these approximations, that we are fitting
a round peg into a square hole. The areas (integrated properties) of a circle
and a square can be matched, but their perimeters (differential properties)
will remain stubbornly different.

1.5
1.5.1

Uniform Electron Gas
Kinetic Energy

Simple systems play an important paradigmatic role in science. For example,
the hydrogen atom is a paradigm for all of atomic physics. In the same way,
the uniform electron gas [24] is a paradigm for solid-state physics, and also for
density functional theory. In this system, the electron density n(r) is uniform
or constant over space, and thus the electron number is infinite. The negative
charge of the electrons is neutralized by a rigid uniform positive background.
We could imagine creating such a system by starting with a simple metal,
regarded as a perfect crystal of valence electrons and ions, and then smearing

out the ions to make the uniform background of positive charge. In fact, the
simple metal sodium is physically very much like a uniform electron gas.
We begin by evaluating the non-interacting kinetic energy (this section)
and exchange energy (next section) per electron for a spin-unpolarized electron gas of uniform density n. The corresponding energies for the spinpolarized case can then be found from (1.126) and (1.127).
By symmetry, the Kohn-Sham potential vs (r) must be uniform or constant, and we take it to be zero. We impose boundary conditions within a
cube of volume V → ∞, i.e., we require that the orbitals repeat from one face
of the cube to its opposite face. (Presumably any choice of boundary conditions would give the same answer
as V → ∞.) The Kohn-Sham orbitals are
√
then plane waves exp(ik · r)/ V, with momenta or wavevectors k and energies k 2 /2. The number of orbitals of both spins in a volume d3 k of wavevector
space is 2[V/(2π)3 ]d3 k, by an elementary geometrical argument [53].
Let N = nV be the number of electrons in volume V. These electrons
occupy the N lowest Kohn-Sham spin orbitals, i.e., those with k < kF :
θ(kF − k) = 2

N =2
k

V
(2π)3

kF
0

dk 4πk 2 = V

kF3
,
3π 2

(1.132)